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class="hands right"></div></div></div></div><div id="container"><header id="header" itemscope itemtype="http://schema.org/WPHeader"><div class="inner"><div id="brand"><div class="pjax"><h1 itemprop="name headline">寒假冬令营DAY3</h1><div class="meta"><span class="item" title="创建时间：2021-01-20 15:01:54"><span class="icon"><i class="ic i-calendar"></i> </span><span class="text">发表于</span> <time itemprop="dateCreated datePublished" datetime="2021-01-20T15:01:54+08:00">2021-01-20</time></span></div></div></div><nav id="nav"><div class="inner"><div class="toggle"><div class="lines" aria-label="切换导航栏"><span class="line"></span> <span class="line"></span> <span class="line"></span></div></div><ul class="menu"><li class="item title"><a href="/" rel="start">Ling Yunchi</a></li></ul><ul class="right"><li class="item theme"><i class="ic i-sun"></i></li><li class="item search"><i class="ic i-search"></i></li></ul></div></nav></div><div id="imgs" class="pjax"><ul><li class="item" 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content="QWQ"></span><div class="body md" itemprop="articleBody"><p>ICPC 寒假冬令营 day3<br><s>不要问我为什么没有前两天，菜鸡摸了两天了 qwqq</s></p><h1 id="数学初步"><a class="anchor" href="#数学初步">#</a> 数学初步</h1><h2 id="几何初步"><a class="anchor" href="#几何初步">#</a> 几何初步</h2><p>用<s>初中</s>高中几何知识做数学题 qwqq</p><h2 id="欧几里得算法扩展欧几里得算法"><a class="anchor" href="#欧几里得算法扩展欧几里得算法">#</a> 欧几里得算法，扩展欧几里得算法</h2><h3 id="欧几里得算法"><a class="anchor" href="#欧几里得算法">#</a> 欧几里得算法</h3><p>用于求整数 a,b 的最大公约数</p><p>公式:</p><p>对于<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>a</mi><mo>&gt;</mo><mi>b</mi><mo separator="true">,</mo><mi>r</mi><mo>=</mo><mi>a</mi><mtext></mtext><mi>m</mi><mi>o</mi><mi>d</mi><mtext></mtext><mi>b</mi></mrow><annotation encoding="application/x-tex">a&gt;b,r=a\,mod\,b</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:.5782em;vertical-align:-.0391em"></span><span class="mord mathnormal">a</span><span class="mspace" style="margin-right:.2777777777777778em"></span><span class="mrel">&gt;</span><span class="mspace" style="margin-right:.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:.8888799999999999em;vertical-align:-.19444em"></span><span class="mord mathnormal">b</span><span class="mpunct">,</span><span class="mspace" style="margin-right:.16666666666666666em"></span><span class="mord mathnormal" style="margin-right:.02778em">r</span><span class="mspace" style="margin-right:.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:.69444em;vertical-align:0"></span><span class="mord mathnormal">a</span><span class="mspace" style="margin-right:.16666666666666666em"></span><span class="mord mathnormal">m</span><span class="mord mathnormal">o</span><span class="mord mathnormal">d</span><span class="mspace" style="margin-right:.16666666666666666em"></span><span class="mord mathnormal">b</span></span></span></span></p><p><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>G</mi><mi>C</mi><mi>D</mi><mo stretchy="false">(</mo><mi>a</mi><mo separator="true">,</mo><mi>b</mi><mo stretchy="false">)</mo><mo>=</mo><mrow><mo fence="true">{</mo><mtable rowspacing="0.15999999999999992em" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mi>b</mi></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mi>r</mi><mo>=</mo><mn>0</mn></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mi>G</mi><mi>C</mi><mi>D</mi><mo stretchy="false">(</mo><mi>b</mi><mo separator="true">,</mo><mi>a</mi><mtext></mtext><mi>m</mi><mi>o</mi><mi>d</mi><mtext></mtext><mi>b</mi><mo stretchy="false">)</mo></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mi>r</mi><mo mathvariant="normal">≠</mo><mn>0</mn></mrow></mstyle></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex">GCD(a,b)=\left\{\begin{matrix} b &amp; r=0\\ GCD(b,a\,mod\,b) &amp; r\neq 0 \end{matrix}\right.</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-.25em"></span><span class="mord mathnormal">G</span><span class="mord mathnormal" style="margin-right:.07153em">C</span><span class="mord mathnormal" style="margin-right:.02778em">D</span><span class="mopen">(</span><span class="mord mathnormal">a</span><span class="mpunct">,</span><span class="mspace" style="margin-right:.16666666666666666em"></span><span class="mord mathnormal">b</span><span class="mclose">)</span><span class="mspace" style="margin-right:.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:2.40003em;vertical-align:-.95003em"></span><span class="minner"><span class="mopen delimcenter" style="top:0"><span class="delimsizing size3">{</span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.45em"><span style="top:-3.61em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathnormal">b</span></span></span><span style="top:-2.4099999999999997em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathnormal">G</span><span class="mord mathnormal" style="margin-right:.07153em">C</span><span class="mord mathnormal" style="margin-right:.02778em">D</span><span class="mopen">(</span><span class="mord mathnormal">b</span><span class="mpunct">,</span><span class="mspace" style="margin-right:.16666666666666666em"></span><span class="mord mathnormal">a</span><span class="mspace" style="margin-right:.16666666666666666em"></span><span class="mord mathnormal">m</span><span class="mord mathnormal">o</span><span class="mord mathnormal">d</span><span class="mspace" style="margin-right:.16666666666666666em"></span><span class="mord mathnormal">b</span><span class="mclose">)</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:.9500000000000004em"><span></span></span></span></span></span><span class="arraycolsep" style="width:.5em"></span><span class="arraycolsep" style="width:.5em"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.45em"><span style="top:-3.61em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathnormal" style="margin-right:.02778em">r</span><span class="mspace" style="margin-right:.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:.2777777777777778em"></span><span class="mord">0</span></span></span><span style="top:-2.4099999999999997em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathnormal" style="margin-right:.02778em">r</span><span class="mspace" style="margin-right:.2777777777777778em"></span><span class="mrel"><span class="mrel"><span class="mord vbox"><span class="thinbox"><span class="rlap"><span class="strut" style="height:.8888799999999999em;vertical-align:-.19444em"></span><span class="inner"><span class="mrel"></span></span><span class="fix"></span></span></span></span></span><span class="mrel">=</span></span><span class="mspace" style="margin-right:.2777777777777778em"></span><span class="mord">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:.9500000000000004em"><span></span></span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></p><h3 id="拓展欧几里得"><a class="anchor" href="#拓展欧几里得">#</a> 拓展欧几里得</h3><p>对于不完全为 0 的非负整数 a，b，gcd（a，b）表示 a，b 的最大公约数，必然存在整数对 x，y ，使得 gcd（a，b）=a<em>x+b</em>y。</p><p><strong>由贝祖定理得</strong>：任意两个整数 a,b, 最大公约数为 d=gcd (a,b), 那么对于任意的整数 x,y，ax+by=m, 构成的 m 一定是 d 的整数倍 (即 m% d=0)</p><p>证明:</p><p>设 a&gt;b。<br>显然当 b=0，gcd（a，b）=a。此时 x=1，y=0；<br>当 ab!=0 时<br>设 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>a</mi><msub><mi>x</mi><mn>1</mn></msub><mo>+</mo><mi>b</mi><msub><mi>y</mi><mn>1</mn></msub><mo>=</mo><mi>g</mi><mi>c</mi><mi>d</mi><mo stretchy="false">(</mo><mi>a</mi><mo separator="true">,</mo><mi>b</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">ax_1+by_1=gcd(a,b)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:.73333em;vertical-align:-.15em"></span><span class="mord mathnormal">a</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0;margin-right:.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:.8888799999999999em;vertical-align:-.19444em"></span><span class="mord mathnormal">b</span><span class="mord"><span class="mord mathnormal" style="margin-right:.03588em">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-.03588em;margin-right:.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-.25em"></span><span class="mord mathnormal" style="margin-right:.03588em">g</span><span class="mord mathnormal">c</span><span class="mord mathnormal">d</span><span class="mopen">(</span><span class="mord mathnormal">a</span><span class="mpunct">,</span><span class="mspace" style="margin-right:.16666666666666666em"></span><span class="mord mathnormal">b</span><span class="mclose">)</span></span></span></span>;<br><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>b</mi><msub><mi>x</mi><mn>2</mn></msub><mo>+</mo><mo stretchy="false">(</mo><mi>a</mi><mtext></mtext><mi>m</mi><mi>o</mi><mi>d</mi><mtext></mtext><mi>b</mi><mo stretchy="false">)</mo><msub><mi>y</mi><mn>2</mn></msub><mo>=</mo><mi>g</mi><mi>c</mi><mi>d</mi><mo stretchy="false">(</mo><mi>b</mi><mo separator="true">,</mo><mi>a</mi><mtext></mtext><mi>m</mi><mi>o</mi><mi>d</mi><mtext></mtext><mi>b</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">bx_2+(a\,mod\,b)y_2=gcd(b,a\,mod\,b)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:.84444em;vertical-align:-.15em"></span><span class="mord mathnormal">b</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0;margin-right:.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-.25em"></span><span class="mopen">(</span><span class="mord mathnormal">a</span><span class="mspace" style="margin-right:.16666666666666666em"></span><span class="mord mathnormal">m</span><span class="mord mathnormal">o</span><span class="mord mathnormal">d</span><span class="mspace" style="margin-right:.16666666666666666em"></span><span class="mord mathnormal">b</span><span class="mclose">)</span><span class="mord"><span class="mord mathnormal" style="margin-right:.03588em">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-.03588em;margin-right:.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-.25em"></span><span class="mord mathnormal" style="margin-right:.03588em">g</span><span class="mord mathnormal">c</span><span class="mord mathnormal">d</span><span class="mopen">(</span><span class="mord mathnormal">b</span><span class="mpunct">,</span><span class="mspace" style="margin-right:.16666666666666666em"></span><span class="mord mathnormal">a</span><span class="mspace" style="margin-right:.16666666666666666em"></span><span class="mord mathnormal">m</span><span class="mord mathnormal">o</span><span class="mord mathnormal">d</span><span class="mspace" style="margin-right:.16666666666666666em"></span><span class="mord mathnormal">b</span><span class="mclose">)</span></span></span></span>;<br>根据朴素的欧几里德原理有 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>g</mi><mi>c</mi><mi>d</mi><mo stretchy="false">(</mo><mi>a</mi><mo separator="true">,</mo><mi>b</mi><mo stretchy="false">)</mo><mo>=</mo><mi>g</mi><mi>c</mi><mi>d</mi><mo stretchy="false">(</mo><mi>b</mi><mo separator="true">,</mo><mi>a</mi><mtext></mtext><mi>m</mi><mi>o</mi><mi>d</mi><mtext></mtext><mi>b</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">gcd(a,b)=gcd(b,a \,mod\,b)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-.25em"></span><span class="mord mathnormal" style="margin-right:.03588em">g</span><span class="mord mathnormal">c</span><span class="mord mathnormal">d</span><span class="mopen">(</span><span class="mord mathnormal">a</span><span class="mpunct">,</span><span class="mspace" style="margin-right:.16666666666666666em"></span><span class="mord mathnormal">b</span><span class="mclose">)</span><span class="mspace" style="margin-right:.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-.25em"></span><span class="mord mathnormal" style="margin-right:.03588em">g</span><span class="mord mathnormal">c</span><span class="mord mathnormal">d</span><span class="mopen">(</span><span class="mord mathnormal">b</span><span class="mpunct">,</span><span class="mspace" style="margin-right:.16666666666666666em"></span><span class="mord mathnormal">a</span><span class="mspace" style="margin-right:.16666666666666666em"></span><span class="mord mathnormal">m</span><span class="mord mathnormal">o</span><span class="mord mathnormal">d</span><span class="mspace" style="margin-right:.16666666666666666em"></span><span class="mord mathnormal">b</span><span class="mclose">)</span></span></span></span>;<br>则:<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>a</mi><msub><mi>x</mi><mn>1</mn></msub><mo>+</mo><mi>b</mi><msub><mi>y</mi><mn>1</mn></msub><mo>=</mo><mi>b</mi><msub><mi>x</mi><mn>2</mn></msub><mo>+</mo><mo stretchy="false">(</mo><mi>a</mi><mtext></mtext><mi>m</mi><mi>o</mi><mi>d</mi><mtext></mtext><mi>b</mi><mo stretchy="false">)</mo><msub><mi>y</mi><mn>2</mn></msub></mrow><annotation encoding="application/x-tex">ax_1+by_1=bx_2+(a \,mod \,b)y_2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:.73333em;vertical-align:-.15em"></span><span class="mord mathnormal">a</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0;margin-right:.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:.8888799999999999em;vertical-align:-.19444em"></span><span class="mord mathnormal">b</span><span class="mord"><span class="mord mathnormal" style="margin-right:.03588em">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-.03588em;margin-right:.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:.84444em;vertical-align:-.15em"></span><span class="mord mathnormal">b</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0;margin-right:.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-.25em"></span><span class="mopen">(</span><span class="mord mathnormal">a</span><span class="mspace" style="margin-right:.16666666666666666em"></span><span class="mord mathnormal">m</span><span class="mord mathnormal">o</span><span class="mord mathnormal">d</span><span class="mspace" style="margin-right:.16666666666666666em"></span><span class="mord mathnormal">b</span><span class="mclose">)</span><span class="mord"><span class="mord mathnormal" style="margin-right:.03588em">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-.03588em;margin-right:.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:.15em"><span></span></span></span></span></span></span></span></span></span>;<br>即:<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>a</mi><msub><mi>x</mi><mn>1</mn></msub><mo>+</mo><mi>b</mi><msub><mi>y</mi><mn>1</mn></msub><mo>=</mo><mi>b</mi><msub><mi>x</mi><mn>2</mn></msub><mo>+</mo><mo stretchy="false">(</mo><mi>a</mi><mo>−</mo><mrow><mo fence="true">⌊</mo><mfrac><mi>a</mi><mi>b</mi></mfrac><mo fence="true">⌋</mo></mrow><mo>∗</mo><mi>b</mi><mo stretchy="false">)</mo><msub><mi>y</mi><mn>2</mn></msub><mo>=</mo><mi>a</mi><msub><mi>y</mi><mn>2</mn></msub><mo>+</mo><mi>b</mi><msub><mi>x</mi><mn>2</mn></msub><mo>−</mo><mrow><mo fence="true">⌊</mo><mfrac><mi>a</mi><mi>b</mi></mfrac><mo fence="true">⌋</mo></mrow><mo>∗</mo><mi>b</mi><msub><mi>y</mi><mn>2</mn></msub></mrow><annotation encoding="application/x-tex">ax_1+by_1=bx_2+(a-\left \lfloor\frac{a}{b}\right \rfloor*b)y_2=ay_2+bx_2-\left \lfloor\frac{a}{b}\right \rfloor*by_2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:.73333em;vertical-align:-.15em"></span><span class="mord mathnormal">a</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0;margin-right:.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:.8888799999999999em;vertical-align:-.19444em"></span><span class="mord mathnormal">b</span><span class="mord"><span class="mord mathnormal" style="margin-right:.03588em">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-.03588em;margin-right:.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:.84444em;vertical-align:-.15em"></span><span class="mord mathnormal">b</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0;margin-right:.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-.25em"></span><span class="mopen">(</span><span class="mord mathnormal">a</span><span class="mspace" style="margin-right:.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:1.20001em;vertical-align:-.35001em"></span><span class="minner"><span class="mopen delimcenter" style="top:0"><span class="delimsizing size1">⌊</span></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:.695392em"><span style="top:-2.6550000000000002em"><span class="pstrut" style="height:3em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">b</span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:.04em"></span></span><span style="top:-3.394em"><span class="pstrut" style="height:3em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">a</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:.345em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mclose delimcenter" style="top:0"><span class="delimsizing size1">⌋</span></span></span><span class="mspace" style="margin-right:.2222222222222222em"></span><span class="mbin">∗</span><span class="mspace" style="margin-right:.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-.25em"></span><span class="mord mathnormal">b</span><span class="mclose">)</span><span class="mord"><span class="mord mathnormal" style="margin-right:.03588em">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-.03588em;margin-right:.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:.7777700000000001em;vertical-align:-.19444em"></span><span class="mord mathnormal">a</span><span class="mord"><span class="mord mathnormal" style="margin-right:.03588em">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-.03588em;margin-right:.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:.84444em;vertical-align:-.15em"></span><span class="mord mathnormal">b</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0;margin-right:.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:1.20001em;vertical-align:-.35001em"></span><span class="minner"><span class="mopen delimcenter" style="top:0"><span class="delimsizing size1">⌊</span></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:.695392em"><span style="top:-2.6550000000000002em"><span class="pstrut" style="height:3em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">b</span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:.04em"></span></span><span style="top:-3.394em"><span class="pstrut" style="height:3em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">a</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:.345em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mclose delimcenter" style="top:0"><span class="delimsizing size1">⌋</span></span></span><span class="mspace" style="margin-right:.2222222222222222em"></span><span class="mbin">∗</span><span class="mspace" style="margin-right:.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:.8888799999999999em;vertical-align:-.19444em"></span><span class="mord mathnormal">b</span><span class="mord"><span class="mord mathnormal" style="margin-right:.03588em">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-.03588em;margin-right:.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:.15em"><span></span></span></span></span></span></span></span></span></span>;<br>根据恒等定理得：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>x</mi><mn>1</mn></msub><mo>=</mo><msub><mi>y</mi><mn>2</mn></msub><mo separator="true">;</mo><msub><mi>y</mi><mn>1</mn></msub><mo>=</mo><msub><mi>x</mi><mn>2</mn></msub><mo>−</mo><mfrac><mi>a</mi><mi>b</mi></mfrac><mo>∗</mo><msub><mi>y</mi><mn>2</mn></msub></mrow><annotation encoding="application/x-tex">x_1=y_2; y_1=x_2-\frac{a}{b}*y_2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:.58056em;vertical-align:-.15em"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0;margin-right:.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:.625em;vertical-align:-.19444em"></span><span class="mord"><span class="mord mathnormal" style="margin-right:.03588em">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-.03588em;margin-right:.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:.15em"><span></span></span></span></span></span></span><span class="mpunct">;</span><span class="mspace" style="margin-right:.16666666666666666em"></span><span class="mord"><span class="mord mathnormal" style="margin-right:.03588em">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-.03588em;margin-right:.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:.73333em;vertical-align:-.15em"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0;margin-right:.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:1.040392em;vertical-align:-.345em"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:.695392em"><span style="top:-2.6550000000000002em"><span class="pstrut" style="height:3em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">b</span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:.04em"></span></span><span style="top:-3.394em"><span class="pstrut" style="height:3em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">a</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:.345em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:.2222222222222222em"></span><span class="mbin">∗</span><span class="mspace" style="margin-right:.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:.625em;vertical-align:-.19444em"></span><span class="mord"><span class="mord mathnormal" style="margin-right:.03588em">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-.03588em;margin-right:.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:.15em"><span></span></span></span></span></span></span></span></span></span>;<br>这样我们就得到了求解 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>x</mi><mn>1</mn></msub><mo separator="true">,</mo><msub><mi>y</mi><mn>1</mn></msub></mrow><annotation encoding="application/x-tex">x_1,y_1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:.625em;vertical-align:-.19444em"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0;margin-right:.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:.15em"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:.16666666666666666em"></span><span class="mord"><span class="mord mathnormal" style="margin-right:.03588em">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-.03588em;margin-right:.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:.15em"><span></span></span></span></span></span></span></span></span></span> 的方法：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>x</mi><mn>1</mn></msub><mtext>，</mtext><msub><mi>y</mi><mn>1</mn></msub></mrow><annotation encoding="application/x-tex">x_1，y_1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:.8777699999999999em;vertical-align:-.19444em"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0;margin-right:.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:.15em"><span></span></span></span></span></span></span><span class="mord cjk_fallback">，</span><span class="mord"><span class="mord mathnormal" style="margin-right:.03588em">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-.03588em;margin-right:.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:.15em"><span></span></span></span></span></span></span></span></span></span> 的值基于 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>x</mi><mn>2</mn></msub><mtext>，</mtext><msub><mi>y</mi><mn>2</mn></msub></mrow><annotation encoding="application/x-tex">x_2，y_2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:.8777699999999999em;vertical-align:-.19444em"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0;margin-right:.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:.15em"><span></span></span></span></span></span></span><span class="mord cjk_fallback">，</span><span class="mord"><span class="mord mathnormal" style="margin-right:.03588em">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-.03588em;margin-right:.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:.15em"><span></span></span></span></span></span></span></span></span></span>.</p><p>a<em>x + b</em>y = gcd (a,b) 是一个不定方程，有多解是一定的，但是只要我们找到一组特殊的解 x0 和 y0 那么，我们就可以用 x0 和 y0 表示出整个不定方程的通解，因为，将 (x0,y0) 代入 ax+by=c，则有</p><p><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>a</mi><mo>∗</mo><msub><mi>x</mi><mn>0</mn></msub><mo>+</mo><mi>b</mi><mo>∗</mo><msub><mi>y</mi><mn>0</mn></msub><mo>=</mo><mi>c</mi></mrow><annotation encoding="application/x-tex">a*x_0+b*y_0=c</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:.46528em;vertical-align:0"></span><span class="mord mathnormal">a</span><span class="mspace" style="margin-right:.2222222222222222em"></span><span class="mbin">∗</span><span class="mspace" style="margin-right:.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:.73333em;vertical-align:-.15em"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0;margin-right:.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:.69444em;vertical-align:0"></span><span class="mord mathnormal">b</span><span class="mspace" style="margin-right:.2222222222222222em"></span><span class="mbin">∗</span><span class="mspace" style="margin-right:.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:.625em;vertical-align:-.19444em"></span><span class="mord"><span class="mord mathnormal" style="margin-right:.03588em">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-.03588em;margin-right:.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:.43056em;vertical-align:0"></span><span class="mord mathnormal">c</span></span></span></span></p><p>通过拆添项，可有：</p><p><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>a</mi><mo>∗</mo><mo stretchy="false">(</mo><msub><mi>x</mi><mn>0</mn></msub><mo>+</mo><mi>b</mi><mo stretchy="false">)</mo><mo>+</mo><mi>b</mi><mo>∗</mo><mo stretchy="false">(</mo><msub><mi>y</mi><mn>0</mn></msub><mo>−</mo><mi>a</mi><mo stretchy="false">)</mo><mo>=</mo><mi>c</mi></mrow><annotation encoding="application/x-tex">a*(x_0+b)+b*(y_0-a)=c</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:.46528em;vertical-align:0"></span><span class="mord mathnormal">a</span><span class="mspace" style="margin-right:.2222222222222222em"></span><span class="mbin">∗</span><span class="mspace" style="margin-right:.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-.25em"></span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0;margin-right:.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-.25em"></span><span class="mord mathnormal">b</span><span class="mclose">)</span><span class="mspace" style="margin-right:.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:.69444em;vertical-align:0"></span><span class="mord mathnormal">b</span><span class="mspace" style="margin-right:.2222222222222222em"></span><span class="mbin">∗</span><span class="mspace" style="margin-right:.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-.25em"></span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal" style="margin-right:.03588em">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-.03588em;margin-right:.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-.25em"></span><span class="mord mathnormal">a</span><span class="mclose">)</span><span class="mspace" style="margin-right:.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:.43056em;vertical-align:0"></span><span class="mord mathnormal">c</span></span></span></span></p><p><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>a</mi><mo>∗</mo><mo stretchy="false">(</mo><msub><mi>x</mi><mn>0</mn></msub><mo>+</mo><mn>2</mn><mo>∗</mo><mi>b</mi><mo stretchy="false">)</mo><mo>+</mo><mi>b</mi><mo>∗</mo><mo stretchy="false">(</mo><msub><mi>y</mi><mn>0</mn></msub><mo>−</mo><mn>2</mn><mo>∗</mo><mi>a</mi><mo stretchy="false">)</mo><mo>=</mo><mi>c</mi></mrow><annotation encoding="application/x-tex">a*(x_0+2*b)+b*(y_0-2*a)=c</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:.46528em;vertical-align:0"></span><span class="mord mathnormal">a</span><span class="mspace" style="margin-right:.2222222222222222em"></span><span class="mbin">∗</span><span class="mspace" style="margin-right:.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-.25em"></span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0;margin-right:.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:.64444em;vertical-align:0"></span><span class="mord">2</span><span class="mspace" style="margin-right:.2222222222222222em"></span><span class="mbin">∗</span><span class="mspace" style="margin-right:.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-.25em"></span><span class="mord mathnormal">b</span><span class="mclose">)</span><span class="mspace" style="margin-right:.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:.69444em;vertical-align:0"></span><span class="mord mathnormal">b</span><span class="mspace" style="margin-right:.2222222222222222em"></span><span class="mbin">∗</span><span class="mspace" style="margin-right:.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-.25em"></span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal" style="margin-right:.03588em">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-.03588em;margin-right:.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:.64444em;vertical-align:0"></span><span class="mord">2</span><span class="mspace" style="margin-right:.2222222222222222em"></span><span class="mbin">∗</span><span class="mspace" style="margin-right:.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-.25em"></span><span class="mord mathnormal">a</span><span class="mclose">)</span><span class="mspace" style="margin-right:.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:.43056em;vertical-align:0"></span><span class="mord mathnormal">c</span></span></span></span></p><p><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>a</mi><mo>∗</mo><mo stretchy="false">(</mo><msub><mi>x</mi><mn>0</mn></msub><mo>+</mo><mn>3</mn><mo>∗</mo><mi>b</mi><mo stretchy="false">)</mo><mo>+</mo><mi>b</mi><mo>∗</mo><mo stretchy="false">(</mo><msub><mi>y</mi><mn>0</mn></msub><mo>−</mo><mn>3</mn><mo>∗</mo><mi>a</mi><mo stretchy="false">)</mo><mo>=</mo><mi>c</mi></mrow><annotation encoding="application/x-tex">a*(x_0+3*b)+b*(y_0-3*a)=c</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:.46528em;vertical-align:0"></span><span class="mord mathnormal">a</span><span class="mspace" style="margin-right:.2222222222222222em"></span><span class="mbin">∗</span><span class="mspace" style="margin-right:.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-.25em"></span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0;margin-right:.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:.64444em;vertical-align:0"></span><span class="mord">3</span><span class="mspace" style="margin-right:.2222222222222222em"></span><span class="mbin">∗</span><span class="mspace" style="margin-right:.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-.25em"></span><span class="mord mathnormal">b</span><span class="mclose">)</span><span class="mspace" style="margin-right:.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:.69444em;vertical-align:0"></span><span class="mord mathnormal">b</span><span class="mspace" style="margin-right:.2222222222222222em"></span><span class="mbin">∗</span><span class="mspace" style="margin-right:.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-.25em"></span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal" style="margin-right:.03588em">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-.03588em;margin-right:.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:.64444em;vertical-align:0"></span><span class="mord">3</span><span class="mspace" style="margin-right:.2222222222222222em"></span><span class="mbin">∗</span><span class="mspace" style="margin-right:.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-.25em"></span><span class="mord mathnormal">a</span><span class="mclose">)</span><span class="mspace" style="margin-right:.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:.43056em;vertical-align:0"></span><span class="mord mathnormal">c</span></span></span></span></p><p>……</p><p><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>a</mi><mo>∗</mo><mo stretchy="false">(</mo><msub><mi>x</mi><mn>0</mn></msub><mo>+</mo><mi>k</mi><mo>∗</mo><mi>b</mi><mo stretchy="false">)</mo><mo>+</mo><mi>b</mi><mo>∗</mo><mo stretchy="false">(</mo><msub><mi>y</mi><mn>0</mn></msub><mo>−</mo><mi>k</mi><mo>∗</mo><mi>a</mi><mo stretchy="false">)</mo><mo>=</mo><mi>c</mi><mo stretchy="false">(</mo><mi>k</mi><mo>∈</mo><mi>Z</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">a*(x_0+k*b)+b*(y_0-k*a)=c (k∈Z)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:.46528em;vertical-align:0"></span><span class="mord mathnormal">a</span><span class="mspace" style="margin-right:.2222222222222222em"></span><span class="mbin">∗</span><span class="mspace" style="margin-right:.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-.25em"></span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0;margin-right:.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:.69444em;vertical-align:0"></span><span class="mord mathnormal" style="margin-right:.03148em">k</span><span class="mspace" style="margin-right:.2222222222222222em"></span><span class="mbin">∗</span><span class="mspace" style="margin-right:.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-.25em"></span><span class="mord mathnormal">b</span><span class="mclose">)</span><span class="mspace" style="margin-right:.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:.69444em;vertical-align:0"></span><span class="mord mathnormal">b</span><span class="mspace" style="margin-right:.2222222222222222em"></span><span class="mbin">∗</span><span class="mspace" style="margin-right:.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-.25em"></span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal" style="margin-right:.03588em">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-.03588em;margin-right:.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:.69444em;vertical-align:0"></span><span class="mord mathnormal" style="margin-right:.03148em">k</span><span class="mspace" style="margin-right:.2222222222222222em"></span><span class="mbin">∗</span><span class="mspace" style="margin-right:.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-.25em"></span><span class="mord mathnormal">a</span><span class="mclose">)</span><span class="mspace" style="margin-right:.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-.25em"></span><span class="mord mathnormal">c</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:.03148em">k</span><span class="mspace" style="margin-right:.2777777777777778em"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-.25em"></span><span class="mord mathnormal" style="margin-right:.07153em">Z</span><span class="mclose">)</span></span></span></span></p><p>又因为我们已知:</p><p>$a<em>X + b</em>Y = gcd(a,b) $</p><p gcd(a,b)="">两边同时乘上\frac{c}</p><p>得到:<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>a</mi><mo>∗</mo><mfrac><mrow><mi>c</mi><mi>X</mi></mrow><mrow><mi>g</mi><mi>c</mi><mi>d</mi><mo stretchy="false">(</mo><mi>a</mi><mo separator="true">,</mo><mi>b</mi><mo stretchy="false">)</mo></mrow></mfrac><mo>+</mo><mi>b</mi><mo>∗</mo><mfrac><mrow><mi>c</mi><mi>Y</mi></mrow><mrow><mi>g</mi><mi>c</mi><mi>d</mi><mo stretchy="false">(</mo><mi>a</mi><mo separator="true">,</mo><mi>b</mi><mo stretchy="false">)</mo></mrow></mfrac><mo>=</mo><mi>c</mi></mrow><annotation encoding="application/x-tex">a*\frac{cX}{gcd(a,b)}+b*\frac{cY}{gcd(a,b)}=c</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:.46528em;vertical-align:0"></span><span class="mord mathnormal">a</span><span class="mspace" style="margin-right:.2222222222222222em"></span><span class="mbin">∗</span><span class="mspace" style="margin-right:.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:1.392331em;vertical-align:-.52em"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:.872331em"><span style="top:-2.655em"><span class="pstrut" style="height:3em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:.03588em">g</span><span class="mord mathnormal mtight">c</span><span class="mord mathnormal mtight">d</span><span class="mopen mtight">(</span><span class="mord mathnormal mtight">a</span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight">b</span><span class="mclose mtight">)</span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:.04em"></span></span><span style="top:-3.394em"><span class="pstrut" style="height:3em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">c</span><span class="mord mathnormal mtight" style="margin-right:.07847em">X</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:.52em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:.69444em;vertical-align:0"></span><span class="mord mathnormal">b</span><span class="mspace" style="margin-right:.2222222222222222em"></span><span class="mbin">∗</span><span class="mspace" style="margin-right:.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:1.392331em;vertical-align:-.52em"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:.872331em"><span style="top:-2.655em"><span class="pstrut" style="height:3em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:.03588em">g</span><span class="mord mathnormal mtight">c</span><span class="mord mathnormal mtight">d</span><span class="mopen mtight">(</span><span class="mord mathnormal mtight">a</span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight">b</span><span class="mclose mtight">)</span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:.04em"></span></span><span style="top:-3.394em"><span class="pstrut" style="height:3em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">c</span><span class="mord mathnormal mtight" style="margin-right:.22222em">Y</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:.52em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:.43056em;vertical-align:0"></span><span class="mord mathnormal">c</span></span></span></span></p><p>至此，我们得到了通解的方程</p><p><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi><mo>=</mo><msub><mi>x</mi><mn>0</mn></msub><mo>+</mo><mfrac><mi>b</mi><mrow><mi>g</mi><mi>c</mi><mi>d</mi><mo stretchy="false">(</mo><mi>a</mi><mo separator="true">,</mo><mi>b</mi><mo stretchy="false">)</mo></mrow></mfrac><mo>∗</mo><mi>k</mi></mrow><annotation encoding="application/x-tex">x = x_0 + \frac{b}{gcd(a,b)}*k</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:.43056em;vertical-align:0"></span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:.73333em;vertical-align:-.15em"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0;margin-right:.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:1.400108em;vertical-align:-.52em"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:.8801079999999999em"><span style="top:-2.655em"><span class="pstrut" style="height:3em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:.03588em">g</span><span class="mord mathnormal mtight">c</span><span class="mord mathnormal mtight">d</span><span class="mopen mtight">(</span><span class="mord mathnormal mtight">a</span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight">b</span><span class="mclose mtight">)</span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:.04em"></span></span><span style="top:-3.394em"><span class="pstrut" style="height:3em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">b</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:.52em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:.2222222222222222em"></span><span class="mbin">∗</span><span class="mspace" style="margin-right:.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:.69444em;vertical-align:0"></span><span class="mord mathnormal" style="margin-right:.03148em">k</span></span></span></span></p><p><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>y</mi><mo>=</mo><msub><mi>y</mi><mn>0</mn></msub><mtext>–</mtext><mfrac><mi>a</mi><mrow><mi>g</mi><mi>c</mi><mi>d</mi><mo stretchy="false">(</mo><mi>a</mi><mo separator="true">,</mo><mi>b</mi><mo stretchy="false">)</mo></mrow></mfrac><mo>∗</mo><mi>k</mi><mo stretchy="false">(</mo><mi>k</mi><mo>∈</mo><mi>Z</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">y = y_0 – \frac{a}{gcd(a,b)}*k (k∈Z)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:.625em;vertical-align:-.19444em"></span><span class="mord mathnormal" style="margin-right:.03588em">y</span><span class="mspace" style="margin-right:.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:1.215392em;vertical-align:-.52em"></span><span class="mord"><span class="mord mathnormal" style="margin-right:.03588em">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-.03588em;margin-right:.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:.15em"><span></span></span></span></span></span></span><span class="mord">–</span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:.695392em"><span style="top:-2.655em"><span class="pstrut" style="height:3em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:.03588em">g</span><span class="mord mathnormal mtight">c</span><span class="mord mathnormal mtight">d</span><span class="mopen mtight">(</span><span class="mord mathnormal mtight">a</span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight">b</span><span class="mclose mtight">)</span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:.04em"></span></span><span style="top:-3.394em"><span class="pstrut" style="height:3em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">a</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:.52em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:.2222222222222222em"></span><span class="mbin">∗</span><span class="mspace" style="margin-right:.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-.25em"></span><span class="mord mathnormal" style="margin-right:.03148em">k</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:.03148em">k</span><span class="mspace" style="margin-right:.2777777777777778em"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-.25em"></span><span class="mord mathnormal" style="margin-right:.07153em">Z</span><span class="mclose">)</span></span></span></span></p><p>这样，所有满足 ax+by=c 的可行解都可求出。</p><p>那我们怎么求出一对特解 x、y 呢？</p><p>由上面的证明可知，对于欧几里得算法，当 a 或 b 有一个为零时，令一个数即为所求的 gcd。所以，欧几里得算法的停止状态为：a=gcd，b=0。</p><p>此时，我们为了满足 a<em>x + b</em>y = gcd ，可以得出 x=1，y=0。</p><p>那么最终状态的值我们就都知道了，x=1、y=0、a=gcd、b=0，而递推式也已经给出，那我们完全可以在求 gcd 的同时，在最终状态对 x，y 进行赋值，然后再在递归回溯的时候依次更新 x，y 的值，返回 gcd 的时候，生成的 x，y 就是一对特解。</p><p>特殊形式<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>a</mi><mo>∗</mo><mi>x</mi><mo>+</mo><mi>b</mi><mo>∗</mo><mi>y</mi><mo>=</mo><mi>g</mi><mi>c</mi><mi>d</mi><mo stretchy="false">(</mo><mi>a</mi><mo separator="true">,</mo><mi>b</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">a*x+b*y=gcd(a,b)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:.46528em;vertical-align:0"></span><span class="mord mathnormal">a</span><span class="mspace" style="margin-right:.2222222222222222em"></span><span class="mbin">∗</span><span class="mspace" style="margin-right:.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:.66666em;vertical-align:-.08333em"></span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:.69444em;vertical-align:0"></span><span class="mord mathnormal">b</span><span class="mspace" style="margin-right:.2222222222222222em"></span><span class="mbin">∗</span><span class="mspace" style="margin-right:.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:.625em;vertical-align:-.19444em"></span><span class="mord mathnormal" style="margin-right:.03588em">y</span><span class="mspace" style="margin-right:.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-.25em"></span><span class="mord mathnormal" style="margin-right:.03588em">g</span><span class="mord mathnormal">c</span><span class="mord mathnormal">d</span><span class="mopen">(</span><span class="mord mathnormal">a</span><span class="mpunct">,</span><span class="mspace" style="margin-right:.16666666666666666em"></span><span class="mord mathnormal">b</span><span class="mclose">)</span></span></span></span> 代码实现:</p><pre><code class="language-c++">int exgcd(int a, int b, int&amp; x, int&amp; y)
&#123;
    if (b == 0)
    &#123;
        x = 1; y = 0;
        return a;
    &#125;
    int t = exgcd(b, a % b, x, y);
    int x_0 = x, y_0 = y;
    x = y_0, y = x_0 * (a / b) * y_0;
    return t;
&#125;
</code></pre><h3 id="拓展循环小数转换为分数的方法"><a class="anchor" href="#拓展循环小数转换为分数的方法">#</a> 拓展：循环小数转换为分数的方法</h3><p>设循环小数 m 有 n 个数字，其中<strong>循环节</strong>有 k 个数字，循环节前有 k-n 个非循环数字</p><p><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>m</mi><mo>=</mo><mn>0.</mn><mi>a</mi><mi>b</mi><mtext></mtext><mi>c</mi><mi>d</mi><mi>e</mi><mi mathvariant="normal">.</mi><mi mathvariant="normal">.</mi><mi mathvariant="normal">.</mi></mrow><annotation encoding="application/x-tex">m=0.ab\,cde...</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:.43056em;vertical-align:0"></span><span class="mord mathnormal">m</span><span class="mspace" style="margin-right:.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:.69444em;vertical-align:0"></span><span class="mord">0</span><span class="mord">.</span><span class="mord mathnormal">a</span><span class="mord mathnormal">b</span><span class="mspace" style="margin-right:.16666666666666666em"></span><span class="mord mathnormal">c</span><span class="mord mathnormal">d</span><span class="mord mathnormal">e</span><span class="mord">.</span><span class="mord">.</span><span class="mord">.</span></span></span></span></p><p>循环小数的 n 个数字组成整数 A,n-k 个非循环数字组成整数 B</p><p><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>A</mi><mo>=</mo><mi>a</mi><mi>b</mi><mi>c</mi><mi>d</mi><mi>e</mi><mo separator="true">,</mo><mi>B</mi><mo>=</mo><mi>a</mi><mi>b</mi></mrow><annotation encoding="application/x-tex">A=abcde,B=ab</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:.68333em;vertical-align:0"></span><span class="mord mathnormal">A</span><span class="mspace" style="margin-right:.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:.8888799999999999em;vertical-align:-.19444em"></span><span class="mord mathnormal">a</span><span class="mord mathnormal">b</span><span class="mord mathnormal">c</span><span class="mord mathnormal">d</span><span class="mord mathnormal">e</span><span class="mpunct">,</span><span class="mspace" style="margin-right:.16666666666666666em"></span><span class="mord mathnormal" style="margin-right:.05017em">B</span><span class="mspace" style="margin-right:.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:.69444em;vertical-align:0"></span><span class="mord mathnormal">a</span><span class="mord mathnormal">b</span></span></span></span></p><p>分数的分母为 k 个 9, 后面补 n-k 个 0; 分数的分子为 A-B</p><p 99900="">m=\frac{abcde-ab}{99900}=\frac{ab*999+cde}{99900}=0.ab+\frac{cde}</p><p>qwqq</p><h2 id="概率论初步"><a class="anchor" href="#概率论初步">#</a> 概率论初步</h2><p>高中概率统计<strong>乘法加法</strong>原理，求解概率问题 qwqq</p><h2 id="微积分初步"><a class="anchor" href="#微积分初步">#</a> 微积分初步</h2><p><s>没讲</s> 见 day4 qwqq</p><h2 id="矩阵计算"><a class="anchor" href="#矩阵计算">#</a> 矩阵计算</h2><p><s>没讲</s> 见 day4 qwqq</p></div><footer><div class="meta"><span class="item"><span class="icon"><i class="ic i-calendar-check"></i> </span><span class="text">更新于</span> <time title="修改时间：2021-01-21 16:53:03" itemprop="dateModified" datetime="2021-01-21T16:53:03+08:00">2021-01-21</time></span></div><div class="reward"><button><i class="ic i-heartbeat"></i> 赞赏</button><p>请我喝[茶]~(￣▽￣)~*</p><div id="qr"><div><img data-src="/images/wechatpay.png" alt="Ling yunchi 微信支付"><p>微信支付</p></div><div><img data-src="/images/alipay.png" alt="Ling yunchi 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title="快速读入函数"><span class="type">上一篇</span> <span class="category"><i class="ic i-flag"></i></span><h3>快速读入函数</h3></a></div><div class="item right"><a href="/2021/01/21/java%E9%9A%8F%E6%89%8B%E7%AC%94%E8%AE%B0/" itemprop="url" rel="next" data-background-image="https:&#x2F;&#x2F;tva4.sinaimg.cn&#x2F;mw690&#x2F;6833939bly1gicli9lfebj20zk0m84qp.jpg" title="java随手笔记"><span class="type">下一篇</span> <span class="category"><i class="ic i-flag"></i></span><h3>java随手笔记</h3></a></div></div><div class="wrap" id="comments"></div></div><div id="sidebar"><div class="inner"><div class="panels"><div class="inner"><div class="contents panel pjax" data-title="文章目录"><ol class="toc"><li class="toc-item toc-level-1"><a class="toc-link" href="#%E6%95%B0%E5%AD%A6%E5%88%9D%E6%AD%A5"><span class="toc-number">1.</span> <span class="toc-text">数学初步</span></a><ol class="toc-child"><li class="toc-item toc-level-2"><a class="toc-link" href="#%E5%87%A0%E4%BD%95%E5%88%9D%E6%AD%A5"><span class="toc-number">1.1.</span> <span class="toc-text">几何初步</span></a></li><li class="toc-item toc-level-2"><a class="toc-link" href="#%E6%AC%A7%E5%87%A0%E9%87%8C%E5%BE%97%E7%AE%97%E6%B3%95%E6%89%A9%E5%B1%95%E6%AC%A7%E5%87%A0%E9%87%8C%E5%BE%97%E7%AE%97%E6%B3%95"><span class="toc-number">1.2.</span> <span class="toc-text">欧几里得算法，扩展欧几里得算法</span></a><ol class="toc-child"><li class="toc-item toc-level-3"><a class="toc-link" href="#%E6%AC%A7%E5%87%A0%E9%87%8C%E5%BE%97%E7%AE%97%E6%B3%95"><span class="toc-number">1.2.1.</span> <span class="toc-text">欧几里得算法</span></a></li><li class="toc-item toc-level-3"><a class="toc-link" href="#%E6%8B%93%E5%B1%95%E6%AC%A7%E5%87%A0%E9%87%8C%E5%BE%97"><span class="toc-number">1.2.2.</span> <span class="toc-text">拓展欧几里得</span></a></li><li class="toc-item toc-level-3"><a class="toc-link" href="#%E6%8B%93%E5%B1%95%E5%BE%AA%E7%8E%AF%E5%B0%8F%E6%95%B0%E8%BD%AC%E6%8D%A2%E4%B8%BA%E5%88%86%E6%95%B0%E7%9A%84%E6%96%B9%E6%B3%95"><span class="toc-number">1.2.3.</span> <span class="toc-text">拓展：循环小数转换为分数的方法</span></a></li></ol></li><li class="toc-item toc-level-2"><a class="toc-link" href="#%E6%A6%82%E7%8E%87%E8%AE%BA%E5%88%9D%E6%AD%A5"><span class="toc-number">1.3.</span> <span class="toc-text">概率论初步</span></a></li><li class="toc-item toc-level-2"><a class="toc-link" href="#%E5%BE%AE%E7%A7%AF%E5%88%86%E5%88%9D%E6%AD%A5"><span class="toc-number">1.4.</span> <span class="toc-text">微积分初步</span></a></li><li class="toc-item toc-level-2"><a class="toc-link" href="#%E7%9F%A9%E9%98%B5%E8%AE%A1%E7%AE%97"><span class="toc-number">1.5.</span> <span class="toc-text">矩阵计算</span></a></li></ol></li></ol></div><div class="related panel pjax" data-title="系列文章"></div><div class="overview panel" data-title="站点概览"><div class="author" itemprop="author" itemscope itemtype="http://schema.org/Person"><img class="image" itemprop="image" alt="Ling yunchi" data-src="/images/avatar.jpg"><p class="name" itemprop="name">Ling yunchi</p><div class="description" 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